Calculating the Nominal Sensitivity of NIRSpec Peter Jakobsen Issue 1.1 – 17 September, 2003

1) Introduction The purpose of this note is to establish a framework for predicting the expected limiting radiometric sensitivity of NIRSpec in its primary observing modes. The conventional expression for calculating the signal-to-noise ratio of one-dimensional spectra extracted from an imaging detector array is derived, and the various parameters of this statistical model are tied to the nominal performance parameters of NIRSpec and the JWST observatory. This methodology is then used to calculate ‘bottom up’ numbers for the expected NIRSpec performance that have been provided to the JWST project as part of the ongoing revision of the formal Level II sensitivity requirements for the JWST instruments. The overall approach employed is closely similar to that used in the earlier memos on this topic by L. Petro, G. Kriss & H. S. Stockman [NGST Sensitivity, STSCI-NGST-TM-2002-0004, 28/03/03] and more recently by P. Lightsey [Sensitivity Requirements Calculation Methodology NGST 03-JWST-0114, 18/04/03] – but is expanded slightly and tailored to the specifics of NIRSpec.

2) The Statistics of Spectral Extraction To understand how the signal-to-noise ratio of astronomical spectra are calculated, it is instructional to first consider how one-dimensional spectra are extracted from two-dimensional detector arrays in practice. [Astronomer readers may safely skip this rather didactic section and jump directly to equation (6)]

B + D S

B˜ + D˜

Figure 1: The projections of two adjacent slits and their corresponding spectra as seen on the detector. The dispersed image of the top slit provides the spectrum of the target superposed on the spectrum of the sky background and the detector dark current. The lower slit provides the spectrum of the adjacent sky background and dark current alone.

As illustrated schematically in Figure 1, the dispersed image of an astronomical source contained within an open shutter of the Micro Shutter Array (MSA) will be imaged on top of the dispersed image of the background sky surrounding the source also entering the shutter. These two photon signals are in turn mixed with the underlying dark current and readout noise of the detector or Focal Plane Array (FPA). Consider a ‘resolution element’ in this spectrum, here for simplicity defined as the area on the detector covered by the projection of the slit at a given wavelength. The raw signal within the resolution element is obtained by summing the signal over the detector pixels spanned by the resolution element
ni =[S + B + D] (1) where S and B are the signals from the source and sky background accumulated during the exposure within the resolution element and D is the signal due to the detector dark current and read-out noise. In the case of the faintest sources where S B + D, the two background signals B and D obviously

1 need to be accurately subtracted from the raw signal to isolate the net object signal S of interest. Since the sky brightness underlying a given target depends on the particulars of the observation and the surrounding target field, the background signal B is seldom known beforehand, but needs to be measured from the exposure itself. In the case of NIRSpec, this will be accomplished by opening one or more shutters containing ’empty sky’ adjacent to the object of interest (normally in the same column) and subtracting the resulting neighboring background spectrum B˜ + D˜ from the raw source spectrum at each wavelength. The resulting measure of the net source signal in the resolution element is then

Sˆ = ni − n˜j =[S + B + D] − [B˜ + D˜] (2)

Since the accumulated photon signals S, B, D, B˜ and D˜ are all stochastic variables (obeying Poisson statistics), Sˆ is clearly also a stochastic variable. Provided the background measured in the adjacent slit is indeed correctly measuring that underlying thesourceinthesenseofE(B˜)=E(B)andE(D˜)=E(D), one has for the average source signal E(Sˆ)

E(Sˆ)=[E(S)+E(B)+E(D)] − [E(B˜)+E(D˜)] = E(S) (3)

In statistical jargon, Sˆ is (by construction) an unbiased estimator of S.SinceS, B, D S˜, B˜ and D˜ are uncorrelated, the statistical variance of Sˆ is simply

Var (Sˆ)=Var (S)+Var (B)+Var (D)+Var (B˜)+Var (D˜) (4)

The extra noise introduced due to fluctuations in the subtracted values of B˜ and D˜ can often be reduced by measuring the background over several nB ≥ 1 different shutters of ’empty sky’ (ideally arranged as a long slit) and subtracting the average measured background from the raw spectrum. In this case:

1 1 Var (B˜)= Var (B)andVar (D˜)= Var (D) (5) nB nB

Combining equations (3), (4) and (5) then gives the general expression for the signal-to-noise ratio of the measurement of S provided by the stochastic variable Sˆ

E(Sˆ) E(S) S/N ≡ √ =    (6) Var (Sˆ) Var (S)+ 1+ 1 (Var (B)+Var (D)) nB

Note that in cases where the object spectrum does not completely fill the slit in the spatial direction, the S/N is maximized by confining the extent of the resolution element to the rows on the detector containing the source signal (see discussion in section 4.5 below). Equation (6) is in this case still applicable, provided nB is interpreted as the ratio of the total number of detector pixels used to measure the background to the number of pixels spanned by the resolution element containing the signal. Note also that in practice, because of pixel-to-pixel variations in the detector dark current and quantum efficiency, in order to achieve E(B)=E(B˜), it is usually necessary to first dark-subtract and flat-field the two-dimensional image using auxiliary calibration data before carrying out the extraction (2). Provided these corrections are correct on average and do not introduce a bulk bias, the residual pixel-to-pixel errors in these initial reduction steps may be treated as an additional source of variance on the variables S and B (see below).

3) Linking to Observatory and Instrument Performance Parameters To proceed further, the statistical parameters of equation (6) need to be linked to the source bright- ness and key performance parameters of the NIRSpec instrument and JWST telescope. Since photon

2 statistics are the dominant source of noise, mathematical clutter is reduced if source brightnesses are consistently expressed in photon units (see Appendix B) and the implicit assumption is made that the NIRSpec detector arrays operate such that each detected photon generates exactly one detected electron irrespective of the wavelength (i.e. QE = DQE). The average source signal accumulated in a resolution element in the case of a continuum point source is

E(S)=fλ ∆λAλ sλ λ QE texp (7) where −1 −2 −1 fλ = source continuum flux in units of photons s cm µm ∆λ = bandwidth spanned by one resolution element at wavelength λ in µm 2 Aλ = effective collecting area of OTE at wavelength λ in cm sλ = slit transmission at wavelength λ λ = net optical throughput of NIRSpec (excluding the MSA and FPA) at wavelength λ QE = quantum efficiency of FPA at wavelength λ texp = total integration time of exposure in s

In the case of a narrow (unresolved) emission line source, expression (7) becomes instead

E(S)=fl Aλ sλ λ QE texp (8) where −1 −2 fl = source line flux in units of photons s cm and other parameters as above The corresponding variance of the source signal is given by

 2 1 δα Var (S)=E(S)+E2(S) (9) npix nexp α where npix = the effective number of pixels spanned by the resolution element over which the signal is extracted nexp = texp/tsub = number of (dithered) sub-exposures each of duration tsub making up the total inte- gration time texp (δα/α) = the residual pixel-to-pixel relative rms fluctuations in detector response after flat-fielding

The first term in (9) is the Poisson noise in the source photon signal. The second term represents the additional noise contribution due to residual uncorrected flat-fielding errors. The average sky background signal under the source signal is similarly

E(B)=npix Zλ ∆λ Ωpix Aλ λ QE texp (10) and its variance

 2 1 δα Var (B)=E(B)+E2(B) (11) npix nexp α where −1 −2 Zλ = is the specific intensity of the (Zodiacal and scattered) background in units of photons s cm µm−1 sr−1

3 Ωpix = the solid angle extended by one detector pixel on the sky in sr and the remaining parameters as above. Lastly, the detector noise in the resolution element is given by

 2 2 2 δc Var (D)=nexpnpix (Ctsub + R +(Ctsub) ) (12) c where C = detector dark current in electrons s−1 per pixel R = net detector read noise per sub-exposure in electrons rms per pixel (δc/c) = relative rms error in detector dark noise subtraction and other parameters as above. The first term in (12) is the contribution due to fluctuations in the detector dark current, which are assumed to obey Poisson statistics. The second term is the net read noise achieved per sub-exposure for the read-out scheme employed. The last term represents the additional noise introduced by errors in the dark current subtraction.

Note that, equations (9), (11) and (12) as written implicitly assume that the nexp sub-exposures making up the total integration are spatially dithered by at least one slit width on the detector so that the resolution element samples a different set of detector pixels during each sub-exposure.

4) Nominal Performance Parameters In this section quantitative values for the performance parameters identified above are compiled and traced to the relevant NIRSpec and JWST reference documents. The key requirement documents are:

[1] JWST Mission Requirements Document (Revision C 09/09/03 – JWST-RQMT-000634) [2] NIRSpec Functional and Performance Requirements Document (Issue 1 25/07/03 – ESA-JWST- RQ-0084) [3] ISIM – NIRSpec Detector Sub-System Functional and Performance Requirements Specification (Issue 1 - 25/07/2003 - ESA-JWST-RQ-22) [4] Calibration concept for the JWST Near-Infrared Spectrograph (NIRSpec) (ST-ECF Instrument Science Report JWST 2003-01) [5] NIRSpec Operations Concept (17/06/03 – STScI-JWST-R-2003-003)

Since the objective is to derive conservative Level II sensitivity requirements for NIRSpec having a suit- able margin, the conservative approach agreed to among the three instruments is to set all ‘stretchable’ performance parameters at their minimum requirement values throughout, and then add another 20% to the calculated nominal sensitivities for contingency.

4.1 Telescope Collecting Area and Throughput (Aλ) The geometrical collecting area of the OTE is specified as 25 m2 [1]. The requirement on the optical throughput of the OTE is presently specified formally as >0.88 at λ ≥ 2.0 µmand>0.50 at λ<2.0 µm [1]. The lax requirement below λ<2 µm is in the process of being refined. At the time of writing, the proposal is to adopt the more detailed table of minimum reflectivities given in Table 1 (e- mail from Mark Clampin to SWG of 04/09/03). The effective collecting area of the OTE at wavelength 5 2 λ is therefore assumed to be Aλ = (λ) × 2.5 × 10 cm with (λ) determined through interpolation of Table 1.

4 Table 1: OTE Optical Throughput

λ (µm)  (%)

0.6 50.0 0.7 61.5 1.0 75.0 1.5 82.0 2.0 88.0 >2.0 88.0

4.2 NIRSpec Image Quality at MSA (sλ) The ‘fat MEMS’ NIRSpec design assumes the use of single 200 mas wide and 450 mas high MSA shutters as the slits for all MOS observations [2]. The fraction of the total light collected by the OTE from a given object that passes through an MSA shutter, sλ, is determined by the NIRSpec Point Spread Function (PSF) at the MSA focal plane and the angular extents of the object and the shutter. Since the slit transmission is most readily quantifiable in the case of a well-centered point source, the NIRSpec sensitivity requirements are implicitly formulated for this case. The NIRSpec image quality at the MSA is specified to be diffraction-limited (Strehl ratio ≥ 0.80) at a wavelength of 2.4 µm [2], corresponding to a total rms wavefront error (WFE) of 180 nm or less (OTE plus NIRSpec foreoptics). NGST/Ball have provided the three instrument teams with sample PSFs for the purpose of sensitivity calculations, these are not sufficient to calculate the NIRSpec slit transmission. For this, the full OTE pupil function is required to model the internal NIRSpec image degradation and take into account the significant diffraction loss (beam spread) that occurs at the MSA shutter. Appendix A describes the simulation used to calculate sλ assuming a detailed OTE pupil function closely similar to those used by NGST/Ball for the 18 segment OTE design. The resulting predicted slit transmission as a function of wavelength for an input PSF at the MSA having a total wavefront error of 180 nm is shown in Figure 2, with selected values of sλ listed in Table 2.

Table 2: Slit Transmission sλ 200 mas × 450 mas slit

λ (µm) sλ (%)

1.2 71.3 1.6 71.0 2.0 70.7 2.7 69.7 3.4 66.7 4.5 57.2

5 Figure 2: Slit transmission sλ as a function of wavelength for a well-centered point source. The full curve shows the predicted transmission for a PSF having a total wavefront error of 180 nm at the MSA. The calculation includes diffraction losses at the MSA assuming 20% oversized NIRSpec optics. The dashed curve shows the corresponding transmission for a perfect image.

Table 3:  = 1000 and  = 3000 Spectral Bands

Band λ1(µm) λ2(µm) λB(µm)

I 1.0 1.8 1.3 II 1.7 3.0 2.2 III 2.9 5.0 3.7

4.3 NIRSpec Spectral Resolution (∆λ) Recall that NIRSpec operates at three spectral resolutions:  = 100,  = 1000, and  = 3000 [2]. The two latter modes cover the 1.0 - 5.0 µm region by means of two sets of three diffraction gratings operating in first order. The two sets of  = 1000 and  = 3000 gratings employ a common set of (long pass) order-isolating filters and span the same three overlapping wavelength bands listed in Table 2. To first order, diffraction gratings provide a constant angular dispersion and therefore a constant pro- jected wavelength per resolution element ∆λ. The spectral resolution, ≡λ/∆λ, in the NIRSpec grating modes therefore increases linearly with wavelength and is set to its nominal value at the central wavelength of each band [2]

(λ1 + λ2) ∆λ =  = 1000 and  = 3000 (13) 2 with λ1 and λ2 listed in Table 3. Note that ∆λ as defined here refers to the bandpass sampled by the resolution element, which is nomi- nally two pixels wide on the detector. The dispersive element of the  = 100 mode is a double-pass prism covering the 0.6 - 5.0 µmregion in a single exposure. The detailed dispersion curve of the prism is not yet available, but is specified to

6 provide a spectral resolution between  =50and = 200 at all wavelengths 1.0 - 5.0 µmand∼100 on a ‘best effort’ basis below 1.0 µm [2]. For lack of better information, the spectral resolution in  = 100 mode is therefore set to the nominal value at all wavelengths

∆λ = λ/= 100 0.6 µm ≤ λ ≤ 5.0 µm (14) where ∆λ again refers to the two pixel wide resolution element on the detector.

4.4 NIRSpec Optical Throughput (λ) The throughput of the NIRSpec optical train (excluding the slit transmission at the MSA) in each of the six grating modes is given by  0 λ =[ i(λ)] G b(λ) (15) i where i(λ) = reflectivity or transmission of each mirror or filter (including surface roughness losses) at wave- length λ 0 G =peakefficiencyofthegrating b(λ) = blaze function of the grating at wavelength λ

The blaze function can depend strongly on polarization, but for the groove densities of interest to NIRSpec is expected to be close to the conventional ‘scalar’ function

2 sin (γ) π(λ − λB) b(λ)= where γ = (16) γ2 λ and λB is the blaze wavelength. Rather than specify all the individual components in the product (15), the NIRSpec requirements [2] stipulate the average and minimum values for the total optical throughput for each mode:

 = 1000: Average λ ≥ 0.45; Minimum λ ≥ 0.35  = 3000: Average λ ≥ 0.35; Minimum λ ≥ 0.30 applicable over each of the three bands listed in Table 3. For computational purposes, these requirements are in the following parameterized by

λ = 0b(λ) (17) where 0 =0.49 for the  = 1000 gratings; 0 =0.39 for the  = 3000 gratings, and b(λ) is calculated from (16) using the blaze wavelengths listed in the last column of Table 3 (here set at λB =2λ1λ2/(λ1 + λ2) to balance the throughput across each band). The requirement on the net throughput of the  = 100 mode is [2]

λ > 0.60 over 1.0 µm ≤ λ ≤ 5.0 µm

Hence for the  = 100 mode, λ =0.60 for all wavelegths 0.6 µm ≤ λ ≤ 5.0 µmisassumed.

4.5 NIRSpec Image Quality and Plate Scale at the FPA (npix, Ωpix) The two optical parameters determining the size of the footprint of the resolution element on the NIRSpec detector array are the image quality and plate scale at the detector focal plane. The latter

7 Figure 3: Simulation showing the appearance of the resolution element at a wavelength of 2.5 µm at the two NIRSpec focal planes: i) image of point source centered within a shutter at the MSA (upper left); ii) the corresponding resolution element as it is imaged on to the FPA (upper right); iii) the same image re-sampled to the 100 mas pixel size of the detector (lower left); and iv) the re-sampled image with 5% detector cross-talk added (lower right). A logarithmic lookup table was employed to better bring out the faint wings of the images. parameter is specified to be 100 mas per pixel (regardless of the final physical pixel size) so that the 200 mas wide single MSA shutter slit always projects to two pixels in the dispersion direction (and 4.5 pixels perpendicular to the dispersion) [2]. The pixel solid angle is therefore −13 Ωpix =2.35 × 10 str The NIRSpec image quality at the FPA is specified to be diffraction-limited (Strehl ratio ≥ 0.80) at a wavelength of 3.0 µm [2], corresponding to a total rms wavefront error WFE ≤ 225 nm (OTE plus NIRSpec foreoptics, collimator and camera). Another important parameter affecting the size of the NIRSpec resolution element on the FPA is the detector pixel-to-pixel cross-talk (related to the detector MTF), which can be quite severe in near-IR arrays, especially at the shorter wavelengths. The NIRSpec requirement is that the cross-talk be ≤ 5% for wavelengths 1.0 µm ≤ λ ≤ 5.0 µm [3]. Figure 3 shows a detailed simulation of how the image of a point source at a wavelength of 2.5 µm appears in the two focal planes in the optical train of NIRSpec (Appendix A). The same OTE and NIRSpec foreoptics WFE used to produce Figure 2 was assumed, but with a further 150 nm of third order (collimator and camera) wavefront error added after the MSA to bring the total rms WFE at the detector focal plane to 225 nm. A detector cross-talk value of 5% was assumed in the last step, meaning that 5% of the light imaged on a given pixel is assumed to leak to each of the four flanking pixels and 2% of the light to each of the four diagonal pixels, leaving 72% in the central pixel. It is evident from Figure 3 that NIRSpec severely under-samples the PSF at the FPA. It is important to appreciate, however, that NIRSpec is detector noise limited at most wavelengths and resolutions, and that the 100 mas angular pixel size has been carefully chosen to optimize the sensitivity to faint sources

8 by projecting the signal on as few detector pixels as possible while fulfilling all the optical constraints (cf. Arribas et al. ST-ECF ISR-NGST-2002-01). In defining what pixels constitute a given resolution element, it is in principle necessary to distinguish between the case where one is interested in measuring a continuum flux (equation (7)) or the flux in an unresolved emission line (equation (8)). In the former case, any smearing of the image of the slit in the dispersion direction will contribute to degrading the spectral resolution and/or spectral purity, but not the sensitivity since the signal lost from a resolution element in the spectral direction will be filled in by that from its neighbors. It is in this case customary to consider the resolution element to be by definition two pixels wide in the dispersion direction, in which case it is only the extent of the PSF in the spatial direction that effects the sensitivity. For a resolution element containing a narrow emission line, however, this filling in of the smeared signal from the neighboring pixels does not take place, and the relevant parameter for the line sensitivity is therefore the full extent of the PSF in both the spatial and spectral dimensions. Another subtlety concerns how exactly the spectrum is extracted from the image as in (2). The simplest approach is for each column to carry out a straight summation of the pixel contents over the number of rows along the slit deemed to contain the source signal. However, in cases where the intensity profile of the signal PSF is accurately known or can be measured from the exposure, it is advantageous to extract the signal from the faintest sources by weighting the pixel contents by the normalized PSF shape before summation. In particular, it can be shown that such a weighted extraction technique is optimal in the sense that it achieves the highest possible S/N by extracting the signal from the smallest possible effective footprint
0 2 −1 npix ≥ npix =( φi ) (18) i where the summation extends over all pixels in the resolution element and φi is the fraction of the light contained in pixel i ( i φi = 1). This limit applies to both the two-dimensional case where the summation is carried out over both dimensions of the PSF image and the one-dimensional case where it is only the extent of the PSF in the spatial direction that is of interest. Although it in the case of extended sources showing complex structure is often not practical (or even 0 advantageous in the case of bright sources) to employ weighted extraction, the parameter npix is a useful objective metric summarizing the impact of the final image quality and pixel size on the sensitivity. 0 Figures 4 and 5 show npix calculated from equation (18) as a function of wavelength using same simu- lation used to produce Figure 3. Figure 4 is for the two-dimensional (emission line) case and Figure 5 is the one-dimensional (continuum flux) case where the resolution element is defined to be two pixels wide, and its effective width is determined by first averaging the image in the spectral dimension and then applying (18) to the resulting one-dimensional spatial image profile. 0 At the large angular pixel size of NIRSpec, the value of npix is quite sensitive to where the image peak lands with respect to the pixel grid. Figures 4 and 5 both refer to the worst case situation where the peak coincides with the intersection of four pixels. 0 These simulations show that the values of npix are not dramatically different in the two- and one- dimensional cases, although, as expected, the effect of detector cross-talk is less pronounced in the latter case since the additional image smearing is only counted in one dimension. These simulations, and considering that the vast majority of NIRSpec target will not be point sources but marginally extended faint galaxies, suggest that it is not unduly conservative to assume that for most NIRSpec targets the signal will be extracted over nearly the entire height of the projected MSA slit image. For the purpose of the NIRSpec sensitivity calculations, the value npix = 8 is therefore assumed for all modes and all wavelengths 0.6 µm ≤ λ ≤ 5.0 µm. This conservative assumption may need to be revisited once better information on the OTE and NIRSpec optics and detector cross-talk becomes available.

9 Figure 4: Minimum extent of the resolution element at the FPA in pixels as a function of wavelength for a well-centered point source calculated according to equation (18) in the 0 two-dimensional (emission line) case. The full curves give npix for a PSF having a WFE of 225 nm and a perfect image, both resampled to 100 mas pixels and assuming 5% pixel-to- pixel cross-talk on the detector. The dashed curves show the equivalent cases if there is no detector cross-stalk. Both sets of curves refer to the worst case sampling where the center of the PSF falls near the intersection of four pixels on the detector.

Figure 5: Same as Figure 4 for the one-dimensional (continuum flux) case where the resolu- tion element per definition has a width of two pixels (the projected slit width) in the spectral direction and the width of the resolution element in the spatial direction is calculated from equation (18).

10 4.6 Detector Quantum Efficiency (QE, δα/α) The requirement on the NIRSpec detector quantum efficiency (or rather DQE) is [3]: QE ≥ 0.8forλ ≥ 1.0 µm QE ≥ 0.7forλ<1.0 µm

The requirement on the rms relative pixel-to-pixel variations in response is (δα/α) ≤ 0.10 [3]. The NIRSpec Calibration Plan [4] assumes that these variations will be mapped to (δα/α) ≤ 0.02 based on ground and in orbit calibrations. For the purpose of the sensitivity calculations, a residual flat fielding error of (δα/α)=0.02 is therefore assumed.

4.7 Cosmic Rays and Integration Times (tsub) Near-IR detector arrays are susceptible to Solar and cosmic ray particle hits, which under quiescent Solar conditions are expected to occur at L2 at a rate of  5eventss−1 cm−2. These cosmic ray events constrain the maximum duration of a NIRSpec exposure. For a NIRSpec detector pixel size of 18 µm [3], and assuming that each event affects 4 adjacent pixels, 4 the above particle rate corresponds to an average time between pixel hits of tc =1.5 × 10 s. Since the cosmic ray hit rate is a Poission process, the waiting time until a hit occurs in a given pixel following a detector reset obeys an exponential distribution

P (t ≤ t˜)=1− exp(−t/t˜ c) (19)

Historically, it has always been assumed that NIRSpec exposures will be built up in increments of sub- exposures of tsub = 1000 s duration, corresponding to  6% of all pixels in each sub-exposure having being hit before the array is reset. Happily, the JWST data and telemetry system has since been scoped to allow the NIRSpec FPA to be operated in a ‘time tagged’ continuous non-destructive read mode with an average of up to four reads being downlinked every tsamp = 50 s [5]. This makes NIRSpec much less susceptible to cosmic rays in that it will enable a posteriori ‘up-the-ramp’ time series analysis of the pixel signals to be carried out, up to and probably beyond the first particle hitting each pixel. This mode of operation will also allow a much better trade-off between detector dark current and read noise to be made by allowing the optimal value of tsub to be selected in orbit.

These potential gains notwithstanding, the conservative value of tsub = 1000 s will be assumed for the purpose of the nominal sensitivity calculations, not least since the formal requirements on the detector noise are formulated in terms of this assumption (see section 4.8). A second issue is what impact the cosmic ray hits may have on the effective exposure time given the foreseen NIRSpec FPA read-out scheme. In the pessimistic case where the signal cannot be picked up again after a cosmic ray event, the average exposure time until a given pixel is first hit in a frame running for a time tsub between resets is t¯exp = tc(1 − exp(−tsub/tc)) (20)

4 which for tsub = 1000 s and tc =1.5 × 10 s amounts to t¯sub =968s. In the event that the pixel signal time series are followed throughout the exposure, only samples contain- ing particle hits become garbled (and that only if onboard averaging of reads is done in the tsamp =50s between downlinks), but provided the offset ‘ramp’ is picked up again after the hit, this does not lead to a loss in exposure time. Since the present consensus is that up-the-ramp sampling will be possible across particle hits, shortening of the net exposure time due to cosmic rays will be ignored for the sensitivity calculations, i.e. t¯sub = tsub = 1000 s will be assumed throughout.

11 This assumption may need ot be revisited as more information on the NIRSpec detectors become avail- able. 4.8 Detector Dark Current and Read Noise (C, R, δc/c) The requirement on the noise performance of the NIRSpec FPA is that the combined noise due to dark current and read-out noise be ≤ 6 electrons per pixel for an integration time of tsub = 1000 s and operating in the up-the-ramp read mode described above [3]. For the sensitivity calculations it will be assumed that the NIRSpec detector operates at its maximum allowed noise performance.

2 −1 Ctsub + R = 6 e pixel for tsub = 1000 s

This parameter is one of the most critical for determining the limiting sensitivity of NIRSpec. In the hope of being able to gain in sensitivity by trading off between dark current and read noise by optimizing the value of tsub = 1000 s in orbit, there is also a separate requirement on the dark current alone [3] C ≤ 0.01 e s−1 per pixel 4 The NIRSpec Calibration Concept Document [4] envisages tmax =2× 10 s of (parallel) observing time being devoted to measuring the pixel-to-pixel dark current map twice per year in orbit, in which case the statistical error in the maps will be   δc 1  √ =0.07 (21) c Ctmax this value together with C =0.01 e s−1 per pixel is assumed in the following.

4.09 Background Light Intensity (Zλ,nB) The limiting sensitivity Level I requirements for the JWST instruments refer to directions toward the Ecliptic Poles where the intensity of the Zodiacal Light is close to its minimum value. Values for the brightness of the Zodiacal Light in this direction to be used for S/N calculations have been tabulated by Petro et al. (STSCI-NGST-TM-2002-0004) and by Denis Ebbets (e-mail to the JWST SWG of 05/06/03), who give the results of three slightly different empirical models based on the COBE obser- vations. These data are plotted in Figure 6. In the case of NIRSpec, the following interpolation formula (a revised version of the expression given in an early version of the NICMOS handbook) is adopted for computational convenience

8 −1.8 −8 −1 −2 −1 −1 Zλ =2.0 × 10 λ(µm) +7.0 × 10 Bλ(Tz) photons s cm µm sr (22) where Bλ(Tz) is the black body Planck function at a temperature of Tz = 256 K (in appropriate photon units). The first term in (22) is the contribution due to scattered solar radiation and the second term is the (diluted) thermal emission from the zodiacal dust. It is seen that equation (22) reproduces the three sets of tabular values to within the scatter over the 1.0 µm <λ<2.0 µm region of interest (24% rms residuals). A second potential source of background in addition to the Zodiacal Light is diffuse starlight scattered off the primary and secondary mirrors of the OTE. The Level II requirement on this background component −1 is Iν ≤ 0.013 MJy sr [1], which is significantly below the Zodiacal Light intensity. In accord with with the philosophy of working to a 20% margin, 1.2 times the intensity predicted by equation (22) is assumed for the sensitivity calculations.

4.10 Image Crowding (nB) The final issue that needs to be addressed is over how large an area adjacent to a given object the spectrum of the sky background can be assumed to be measurable over (as per the discussion in Section 2

12 Figure 6: Intensity of the Zodiacal light toward the Ecliptic poles calculated according to the expression (22) compared to the values tabulated by Petro et al. (green squares) and Ebbets (red triangles and blue circles). Also plotted is the assumed level of scattered starlight of 0.013 MJy sr−1 (dashed line). above). This will depend on the nature of the field surrounding the target being observed and how many nearby sources are also being observed during the exposure. The most conservative assumption is that each target will be accompanied by one MSA shutter of nearby blank sky. Since, as discussed in section 4.5 above, the target signals are expected to essentially take up the entire height of its shutter, this corresponds to adopting nB = 1 in equation (6).

5) Predicted Nominal Performance Since the  = 1000 and  = 3000 modes of NIRSpec are first and foremost intended for emission line spectroscopy, the primary sensitivity parameter of interest for these modes is the limiting unresolved 5 line flux detectable at S/N = 10 in a total integration time of texp =10 s. For  = 100 mode the key performance parameter is the limiting continuum flux detectable at S/N =10 4 per resolution element in a total integration time of texp =10 s. Figures 7 through 10 plot the corresponding limiting sensitivities calculated by inserting equations (7) through (12) into (6) and solving for the input photon flux fl or fλ, using the following parameters discussed in the previous section: 5 2 Aλ:2.5 × 10 cm geometrical area times reflectivity listed in Table 1. ∆λ:  =1000and = 3000: equation (13) with bandpasses listed in Table 3  = 100: equation (14) λ:  = 1000: equations (16) & (17) with 0 =0.49 and λB listed in Table 3  = 3000: Same with 0 =0.39  = 100: 0.6 sλ: values in Figure 2 and Table 2 (180 nm WFE PSF at MSA, single 200 mas × 450 mas shutter, 20% over-sized NIRSpec optics) QE:0.8forλ ≥ 1.0 µm 0.7forλ<1.0 µm (δα/α): 0.02 C:0.01es−1 per pixel

13 (δc/c√): 0.07 2 −1 R: Ctsub + R = 6 e pixel for tsub = 1000 s −13 Ωpix:2.35 × 10 str (100 mas pixels) npix: 8 (225 nm WFE at detector, 5% pixel-to-pixel cross-talk) −1 Zλ:1.2× equation (22) plus 0.013 MJy sr nB:1 tsub: 1000 s

Note that the  = 3000 mode sensitivity of Figure 10 was also calculated for an assumed 200 mas × 450 mas slit, and therefore implicitly refers to fixed slit mode rather than integral field mode (the throughput requirement on the latter has not yet been formalized). As expected, the limiting line sensitivity of the severely background limited  = 1000 and  = 3000 modes are comparable, and only differ due to the slightly lower requirement on the optical throughput of the  = 3000 mode (section 4.4). The much larger difference in continuum sensitivity between these two modes reflects the additional factor of three difference in spectral resolution. In contrast,  = 100 mode is nearly sky-limited, with the Zodiacal Light background signal being greater or equal to the net detector background signal over most of the wavelength range of the instrument. Tables 4, 5 and 6 tabulate selected reference values of the calculated nominal limiting sensitivities with 20% margin added for the  = 1000,  = 100 and  = 3000 modes, respectively. These fluxes (two per band in  = 1000 and  = 3000 mode) are also indicated in Figures 7 through 10. At present, the intention is to elevate the  = 1000 λ=2.0 µm line flux entry of Table 4 and the  =100 λ=3.0 µm continuum flux entry of Table 5 as the formal Observatory Level II minimum sensitivity requirements for NIRSpec. If so, the other entries of Tables 4, 5 and 6 should be held as requirements at the instrument requirement level. Converted to official funky units (Appendix B), the proposed Level II requirements become:

NIRSpec shall in  = 1000 mode be capable of measuring the flux in an unresolved emission −22 −2 line of Fl =5.2 × 10 Wm at S/N = 10 from an unresolved point source at an observed 5 wavelength of λ=2 µminanexposuretimeoftexp =10 sorless.

NIRSpec shall in  = 100 mode be capable of measuring a continuum flux of Fν =1.2 × 10−33 Wm−2 Hz−1 at S/N = 10 from an unresolved point source at an observed wavelength 4 of λ=3 µminanexposuretimeoftexp =10 sorless.

Acknowledgements: Santiago Arribas and Torsten B¨oker are thanked for careful readings of earlier versions of this note.

14 Figure 7: Limiting emission line sensitivity of NIRSpec in  = 1000 mode using nominal 5 performance parameters for texp =10 sandS/N = 10. The colors refer to the three gratings of the  = 1000 mode. Also plotted are the selected values with 20% margin added listed in Table 4

Figure 8: As Figure 7, but showing the corresponding limiting continuum sensitivity of 5 NIRSpec in  = 1000 mode, again for texp =10 sandS/N = 10 per resolution element.

15 4 Figure 9: Limiting continuum sensitivity of NIRSpec in  = 100 mode for texp =10 s and S/N = 10 per resolution element. Selected values with 20% margin added are listed in Table 5. The discontinuity at λ =1.0 µm is an artifact of the step-function nature of the requirement on the detector quantum efficiency (4.6).

Figure 10: Limiting emission line sensitivity of NIRSpec in  = 3000 mode using nominal 5 performance parameters for texp =10 sandS/N = 10. Selected values with 20% margin added are listed in Table 6

16 Table 4:  = 1000 Mode Limiting Flux 5 S/N =10,texp =10 s, 20% margin added

−1 −2 λ (µm) Fν (nJy) Fl (erg s cm )

1.2 336 9.8 × 10−19 1.6 451 7.4 × 10−19 2.0 297 5.2 × 10−19 2.7 431 4.2 × 10−19 3.4 312 3.2 × 10−19 4.5 539 3.2 × 10−19

Table 5:  = 100 Mode Limiting Flux 4 S/N =10,texp =10 s, 20% margin added

−1 −2 λ (µm) Fν (nJy) Fl (erg s cm )

1.5 136 2.7 × 10−18 3.0 118 1.2 × 10−18 4.5 192 1.3 × 10−18

Table 6:  = 3000 Mode Limiting Flux 5 S/N =10,texp =10 s, 20% margin added

−1 −2 λ (µm) Fν (nJy) Fl (erg s cm )

1.2 1213 1.2 × 10−18 1.6 1657 9.1 × 10−19 2.0 1081 6.3 × 10−19 2.7 1595 5.1 × 10−19 3.4 1147 3.9 × 10−19 4.5 1935 3.8 × 10−19

17 Appendix A: OTE Image Quality and Optical Simulations As stated in sections 4.2 and 4.5, calculation of the NIRSpec slit transmission and detector PSF footprint requires that the internal aberrations of the NIRSpec optics and the significant diffraction losses at the MSA are both taken into account. This, in turn, requires full knowledge of the amplitude and phase of the image at the two focal planes of NIRSpec, and not just the intensity of the input PSF of the OTE. For this reason it is necessary to simulate the image quality of the entire OTE + NIRSpec optical chain. The calculations presented in this document employ conventional Fourier optics and assume the most recent 655 cm flat-to-flat 18 element telescope design. The telescope is allowed to display third order surface errors on the scale of the full aperture, and arbitrary tip/tilt, piston and surface errors on the scale of the individual segments. High frequency tooling errors are modelled as random Gaussian surface error variations on the spatial sampling scale of the pupil function.

Figure A1: Assumed OTE WFE map used for the slit transmission and PSF footprint calculations of sections 4.2 and 4.5. A flat-to-flat pupil size of 655 cm, an inter-segment gap of 1.5 cm, and a spider width of 10 cm was assumed. The total rms WFE is 150 nm, made up of 130 nm of large scale astigmatism, 70 nm of mid frequency segment tip/tilt, piston and segment figure error, and 28 nm of random high frequency surface error.

The adopted OTE pupil map is shown in Figure A1 and aims to reproduce the ‘150:133,64,28’ telescope model of NGST/Ball (e-mail from Mark Clampin to SWG of 14/08/03). The model has a total WFE of 150 nm rms, made up of 130 nm of large scale astigmatism, 70 nm of combined mid-frequency tip/tilt, piston and segment surface error, and 28 nm of high frequency error. The encircled energy and azimuthally averaged profiles of the corresponding PSF in the OTE focal plane at wavelengths of 1.0 µm and 2.0 µm are shown in Figures A2 and A3, respectively. At 1.0 µmthe predicted encircled energy at 150 mas radius is 71%, just slightly below the 72% of the target NGST/Ball model (which – as is the case with the NGST/Ball model – does not meet the current 75% requirement). More relevant to NIRSpec (given its 200 mas wide slits) is the encircled energy at 100 mas radius, which

18 Figure A2: Encircled energy and azimuthally averaged image profile for the OTE Pupil Function of Figure A1 at a wavelength of 1.0 µm. The encircled energy at 150 mas radius is 71% at this wavelength.

Figure A3: Encircled energy and azimuthally averaged image profile for the OTE Pupil Function of Figure A1 at a wavelength of 2.0 µm. The Strehl ratio is (by construction) 0.8 at this wavelength.

19 is 61% at 1.0 µm; closely similar to the NGST/Ball model. At 2.0 µm, the encircled energy at 100 mas rises to 63%, compared to 62% in the NGST/Ball model. It would appear that the two models are sufficiently close in their predictions to be considered identical. Armed with this WFE map for the OTE, the NIRSpec optical simulation proceeds as follows:

1) Simulate the aberrations in the NIRSpec foreoptics by adding 100 nm of third order coma to the OTE the pupil map, bringing the total WFE 180 nm.

2) Convert the WFE map to a phase map for the wavelength in question and add a constant amplitude flux whose intensity integrates to unity over the OTE pupil.

3) Fast Fourier Transform the resulting pupil image to produce the corresponding PSF in the MSA focal plane.

4) Simulate the MSA by zeroing the image outside the open shutter area.

5) Transform the truncated image back to the pupil plane.

6) Take into account the diffraction losses by zeroing the pupil image falling outside an oversized circular pupil having a diameter equivalent to 1.2 × the flat-to-flat diameter of the OTE.

7) Sum the intensity inside the truncated pupil image to obtain the slit transmission sλ.

8) Simulate the aberrations in the NIRSpec collimator and camera optics by adding 150 nm of mixed 3rd order aberrations to the truncated pupil image to bring the total WFE to 225 nm.

9) Transform to the image plane to produce the corresponding PSF in the FPA focal plane.

10) Rebin the PSF to the 100 mas pixels of the FPA.

11) Simulate the detector cross-talk by convolving the resampled PSF with the diffusion kernel.

12) Re-normalize the resulting PSF and calculate npix per equation (18).

20 Appendix B: Astronomical Flux Units for the Non-Astronomer The JWST project – with its meeting of conventional UV/optical astronomers from the HST community and diehard infrared astronomers from the SIRTF and ISO communities – encounters something of an astronomical culture clash when it comes to unit conventions. Optical astronomers, in deference to their long and illustrious history, still mostly prefer cgs units, −1 −2 −1 and express wavelengths in units of A˚ and continuum fluxes in Fλ using units of erg s cm A˚ . The infrared astronomers (crass newcomers they are) have taken to using SI units and customarily express wavelengths in units of µm and continuum fluxes in Fν in units of Janskys (borrowed from radio astronomy) – or, in the case of JWST, nJy – where 1 µm=104 A=10˚ −4 cm and 1nJy=10−32 erg s−1 cm−2 Hz−1

The conversion between Fλ and Fν in these preferred units (recall λFλ = νFν )is

21 2 −1 −2 −1 Fν [nJy] = 3.34 × 10 λ [µm] Fλ[erg s cm A˚ ]

The conversions to differential photon flux (equation (7)) are

−1 −2 −1 15 −1 −2 −1 fλ[photons s cm µm ]=5.04 × 10 λ[µm] Fλ[erg s cm A˚ ]

−6 −1 −2 −1 1.51 × 10 fλ[photons s cm µm ]= Fν [nJy] λ[µm]

One bridge between the two cultures is the ‘monochromatic’ AB magnitude system, which, although originating in optical astronomy, is defined in terms of Fν

AB =31.43 − 2.5 log(Fν [nJy])

It is important to distinguish between AB magnitudes and the customary finite-band standard magni- tude systems of conventional astronomy (U, B, V, R, I, J, K, etc.). Conversion between these systems is possible, but involves arcane discussions of color corrections and zero point calibrations that are best left to the professionals.

Optical astronomers tend to express line fluxes in units of erg s−1 cm−2, infrared astronomers in W m−2. The conversion is

−1 −2 3 −2 Fl[erg s cm ]=10 Fl[W m ]

The conversion to photon flux (equation (8)) is

−1 −2 11 −1 −2 fl[photons s cm ]=5.04 × 10 λ[µm] Fl[erg s cm ]

21